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A Modification of the Probability Weighted Method of Moments and its Application to Estimate the Financial Return Distribution Tail

100%
Małecka M., Pekasiewicz D.
Statistics in Transition new series
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2014
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vol. 15
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issue 3
495–506
EN
The issue of fitting the tail of the random variable with an unknown distribution plays a pivotal role in finance statistics since it paves the ground for estimation of high quantiles and subsequently offers risk measures. The parametric estimation of fat tails is based on the convergence to the generalized Pareto distribution (GPD). The paper explored the probability weighted method of moments (PWMM) applied to estimation of the GPD parameters. The focus of the study was on the tail index, commonly used to characterize the degree of tail fatness. The PWMM algorithm requires specification of the cdf estimate of the so-called excess variable and depends on the choice of the order of the probability weighted moments. We suggested modification of the PWMM method through the application of the level crossing empirical distribution function. Through the simulation study, the paper investigated statistical properties of the GPD shape parameter estimates with reference to the PWMM algorithm specification. The simulation experiment was designed with the use of fat-tailed distributions with parameters assessed on the basis of the empirical daily data for DJIA index. The results showed that, in comparison to the commonly used cdf formula, the choice of the level crossing empirical distribution function improved the statistical properties of the PWMM estimates. As a complementary analysis, the PWMM tail estimate of DJIA log returns distribution was presented.
2
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Generalized Pareto distribution based on generalized order statistics and associated inference

88%
Malik M. R., Kumar D.
Statistics in Transition new series
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2019
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vol. 20
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issue 3
57-79
EN
In this paper, we have considered the generalized Pareto distribution. Various structural properties of the distribution are derived including (quantile function, explicit expressions for moments, mean deviation, Bonferroni and Lorenz curves and Renyi entropy). We have provided simple explicit expressions and recurrence relations for single and product moments of generalized order statistics from the generalized Pareto distribution. The method of maximum likelihood is adopted for estimating the model parameters. For different parameter settings and sample sizes, the simulation studies are performed and compared to the performance of the generalized Pareto distribution.
3

A Comparative Analysis of Value at Risk Measurement on Emerging Stock Markets: Case of Montenegro

75%
Cerović J., Lipovina-Božović M., Vujošević S.
Business Systems Research Journal
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2015
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vol. 6
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issue 1
36-55
EN
Background: The concept of value at risk gives estimation of the maximum loss of financial position at a given time for a given probability. The motivation for this analysis lies in the desire to devote necessary attention to risks in Montenegro, and to approach to quantifying and managing risk more thoroughly. Objectives: This paper considers adequacy of the most recent approaches for quantifying market risk, especially of methods that are in the basis of extreme value theory, in Montenegrin emerging market before and during the global financial crisis. In particular, the purpose of the paper is to investigate whether extreme value theory outperforms econometric and quantile evaluation of VaR in emerging stock markets such as Montenegrin market. Methods/Approach: Daily return of Montenegrin stock market index MONEX20 is analyzed for the period January, 2004 - February, 2014. Value at Risk results based on GARCH models, quantile estimation and extreme value theory are compared. Results: Results of the empirical analysis show that the assessments of Value at Risk based on extreme value theory outperform econometric and quantile evaluations. Conclusions: It is obvious that econometric evaluations (ARMA(2,0)- GARCH(1,1) and RiskMetrics) proved to be on the lower bound of possible Value at Risk movements. Risk estimation on emerging markets can be focused on methodology using extreme value theory that is more sophisticated as it has been proven to be the most cautious model when dealing with turbulent times and financial turmoil.
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